Green function of the Poisson equation: D = 2, 3, 4

Abstract

We study the Green function of the Poisson equation in two, three and four dimensions.The solution g of the equation ∇→2 g(x→ - x→’) = δ(D) (x→ - x→’) where x→ and x→’ are D-dimensional position vectors, is customarily expanded into radial and angular coordinates. For the two-dimensional case (D = 2),we find a subtle interplay of the necessarily introduced scale L with the radial component of zeromagnetic quantumnumber. For D = 3, thewell-known expressions are briefly recalled; this is done in order to highlight the analogywith the four-dimensional case,where we uncover analogies of the fourdimensional spherical harmonicswith the familiar three-dimensional case. Remarks on theSO(4) symmetry of the hydrogen atomcomplete the investigations.